Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum pseudorandomness bounds, and more
Jonathan Allcock, Joao F. Doriguello, Gábor Ivanyos, Miklos Santha
TL;DR
This work generalises Bell sampling from qubits to qudits across all dimensions $d\ge 2$ by introducing a new unitary $\mathcal{B}_{\mathbf{R}}$ (built from Lagrange's four-square theorem) that maps $|\mathcal{S}\rangle^{\otimes 4}$ to $|\mathcal{S}^*\rangle^{\otimes 4}$ up to a Pauli, enabling robust sampling of stabiliser information via skewed Bell difference sampling. Leveraging this primitive, the authors lift key qubit results to the qudit setting: stabiliser-state learning in $O(n^3)$ time with $O(n)$ samples, a Quantum State Hidden Subgroup analogue that identifies Weyl groups using $O(n/\varepsilon)$ samples, and stabiliser-size property testing with dimension- and prime-factor-aware complexities. They also obtain exponential improvements on pseudorandomness bounds for $t$-doped Clifford circuits (showing no pseudorandom outputs for $t< n/2$) and present tolerant stabiliser testing algorithms valid for all dimensions. The framework interleaves stabiliser theory with a rich algebraic backbone of rings and modules, symplectic products, and Smith normal form, yielding dimension-agnostic tools for learning, testing, and distinguishing stabiliser-structured quantum states. Overall, the paper significantly broadens the reach of Bell sampling techniques to qudits, with implications for quantum learning, circuit characterization, and cryptographic notions of pseudorandomness.
Abstract
Bell sampling is a simple yet powerful tool based on measuring two copies of a quantum state in the Bell basis, and has found applications in a plethora of problems related to stabiliser states and measures of magic. However, it was not known how to generalise the procedure from qubits to $d$-level systems -- qudits -- for all dimensions $d > 2$ in a useful way. Indeed, a prior work of the authors (arXiv'24) showed that the natural extension of Bell sampling to arbitrary dimensions fails to provide meaningful information about the quantum states being measured. In this paper, we overcome the difficulties encountered in previous works and develop a useful generalisation of Bell sampling to qudits of all $d\geq 2$. At the heart of our primitive is a new unitary, based on Lagrange's four-square theorem, that maps four copies of any stabiliser state $|\mathcal{S}\rangle$ to four copies of its complex conjugate $|\mathcal{S}^\ast\rangle$ (up to some Pauli operator), which may be of independent interest. We then demonstrate the utility of our new Bell sampling technique by lifting several known results from qubits to qudits for any $d\geq 2$: 1. Learning stabiliser states in $O(n^3)$ time with $O(n)$ samples; 2. Solving the Hidden Stabiliser Group Problem in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 3. Testing whether $|ψ\rangle$ has stabiliser size at least $d^t$ or is $\varepsilon$-far from all such states in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 4. Clifford circuits with at most $n/2$ single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether $|ψ\rangle$ has stabiliser fidelity at least $1-\varepsilon_1$ or at most $1-\varepsilon_2$ with $O(d^2/\varepsilon_2)$ samples if $\varepsilon_1 = 0$ or $O(d^2/\varepsilon_2^2)$ samples if $\varepsilon_1 = O(d^{-2})$.